Industry practice commonly relies on dynamic sonic-based data to solve for the elastic moduli, which must be converted to static (rock mechanics-based) moduli using empirical dynamic-static transforms. Examples include the method described in WO2009108432.
Despite success in conventional reservoirs, the sonic-based approach has not been accurate or reliable on many non-conventional rock layers, such as shale, mudstone or marl, which are strongly heterogeneous and exhibit ductile behavior. Research has been published attempting to empirically correlate mineralogy to mechanical properties for different rock formation, but have been less than satisfactory. A common intensive rock mechanics analysis is to test numerous rock samples in the laboratory and to study their mechanical variations. However, this approach is not possible in reservoir settings where physical sampling is scarce. A common petrophysical application is to cross-plot dynamic elastic properties versus porosity or some other rock parameter to derive empirical relationships for the field. Micromechanical techniques are often employed in petrophysical applications for two-phase composites, such as solid and pore-space.
In continuum mechanics, the term Eshelby's inclusion problems refers to a set of problems involving ellipsoidal elastic inclusions in an infinite elastic body. An “inclusion” is a region in an infinite homogeneous isotropic elastic medium undergoing a change of shape and size which, but for the constraint imposed by its surroundings (the “matrix”), would be arbitrary homogeneous strain. Analytical solutions to these problems were first devised by John D. Eshelby in 1957. Eshelby found that the resulting elastic field can be found using a “sequence of imaginary cutting, straining and welding operations.” Eshelby's finding that the strain and stress field inside the ellipsoidal inclusion is uniform and has a closed-form solution, regardless of the elastic material properties and initial transformation strain (also called the eigenstrain), has spawned a large amount of work in the mechanics of composites.
Micromechanics is an approach for predicting behaviors of heterogeneous materials. Heterogeneous materials, such as composites, solid foams, polycrystals, or bone, consist of clearly distinguishable constituents (or phases) that show different mechanical and physical material properties. One goal of micromechanics of materials is predicting the response of the heterogeneous material on the basis of the geometries and properties of the individual constituents, which is known as homogenization. Another goal is localization of materials, which attempts to evaluate the local stress and strain fields in the phases for given macroscopic load states, phase properties and phase geometries.
Because most heterogeneous materials show a statistical rather than a deterministic arrangement of the constituents, the methods of micromechanics are typically based on the concept of the representative volume element (RVE). An RVE is understood to be a sub-volume of an inhomogeneous medium that is of sufficient size for providing all geometrical information necessary for obtaining an appropriate homogenized behavior. Most methods in micromechanics of materials are based on continuum mechanics rather than on atomistic approaches such as molecular dynamics.
Currently there are several mechanical solutions for heterogeneous rocks. For example, Single Elastic Inclusion has long been implemented in elasticity calculations. Eshelby's formula leads to the response of a single ellipsoidal elastic inclusion in an elastic whole space to a uniform strain imposed at infinity. In single elastic inclusion, the rock is assumed to be an isotropic and homogeneous elastic medium. However, calculating the external fields to the inclusions can be laborious.
The second is Multiple Elastic Inclusions, which assumes there are infinite number of elastic domains and can be expressed by:
      C    _    =      C    +                  ∑                  α          =          1                N            ⁢                                    f            α                    ⁡                      (                                          C                α                            -              C                        )                          ⁢                  :                ⁢                  J          α                    In which C is the overall elasticity tensor, C and Cα are the elasticity of the matrix and inclusion phases, respectively, and f is the volume fraction of the inclusion phase, Jα is a constant fourth order tensor that relates the average strain of the inclusion. The strain concentration tensor Jα and the relationship (Cα−C) are estimated by using various averaging schemes such as the dilute distribution assumption, the self-consistent method, or the Mori-Tanaka method.
These approaches cannot efficiently model the stress field along the wellbore with minimum mineralogy provided. Thus, a better method is needed in the art.